## Numerical calculation of the spectrum as a function of *v*

**Figure 1.** Numerical calculation of the spectrum as a function of *v*. At *v* = 0, all energy levels plotted have energy E\le 7+\frac{3}{2}. As the spin–orbit strength is increased, the levels split off into groups with increasing radial quantum number *n*. States of equal *j* are labelled with the same colour, with j=\frac{1}{2} given by black, and j=\frac{21}{2} given by the lightest grey. Each Landau level has energy of approximately E_{njm}=(n+\frac{1}{2})+\frac{(j+\frac{1}{2})^{2}}{2v^{2}}-\frac{v^{2}}{2}. As a visual guide, each level is shifted by *v*^{2}/2. Two regimes are clearly identifiable, corresponding to a 3D harmonic oscillator with slight level mixing, and the quasi-1D Landau-level problem. As discussed in the text, the crossover between these two regimes is given by *E* ~ *v*^{2}/2. This crossover is shown by the black line, and the quasi-1D regime is shaded grey.

**Abstract**

We investigate the properties of an atom under the influence of a synthetic three-dimensional spin–orbit coupling (Weyl coupling) in the presence of a harmonic trap. The conservation of total angular momentum provides a numerically efficient scheme for finding the spectrum and eigenfunctions of the system. We show that at large spin–orbit coupling the system undergoes dimensional reduction from three to one dimension at low energies, and the spectrum is approximately Landau level-like. At high energies, the spectrum is approximately given by the three-dimensional isotropic harmonic oscillator. We explore the properties of the ground state in both position and momentum space. We find the ground state has spin textures with oscillations set by the spin–orbit length scale.